Home » Articles posted by John Khan (Page 9)

# Author Archives: John Khan

## On an integral inequality

Let and be a continuous and increasing function. Prove that

Solution

We are applying Chebyshev’s inequality .

and this finishes the exercise.

## An almost zero function

Let be a Riemann integrable function. If forall rationals of the interval then prove that .

Solution

Since is Riemann integrable this means that the set of discontinuities has a zero measure. Wherever is continuous , it’s gonna be zero due to the rationals being dense. Thus, is almost everywhere zero. But then its Lebesgue integral is zero and so is the Riemann integral.

## The matrix is symmetric

Let be a square matrix and for that it holds that

Prove that is symmetric.

Solution

Let be an square matrix over a field such that

(1)

Taking transposed matrices back at we get that

and thus

(2)

On the other hand it holds that

since

Of course it holds that if a matrix is symmetric or antisymmetric and then . The proof is left as an exercise to the reader.

We can safely conclude using the above observations that for our matrix it holds that

(3)

But then for the matrix it holds that

and since the matrix is antisymmetric we conclude that and thus . Hence the result.

## On permutation

For any permutation define its displacement  as

What is greater: the sum of displacements of even permutations or the sum of displacements of odd permutations? The answer may depend on .

Solution

The sum of over the even permutations minus the one over the odd permutations is the determinant of the matrix with entries and this determinant is known to be

## Equal determinants

Let that are diagonizable in . If and   then prove that

Solution

The key point here is that the matrices are simultaneously diagonisable. Thus , there exists an invertible matrix and diagonisable matrices such that

Hence

where are the diagonial elements of and of . According to we have that

But , so and finally we conclude that: